Following Positive proportion of logarithmic gaps between consecutive primes let for given $\lambda$, $\alpha$ and for any $x$ all positive the quantities $S^{-}_{\lambda,\alpha}(x):=\#\{p_{n+1}\leqslant x\colon p_{n+1}-p_{n}\leqslant\lambda\log^{\alpha}x\}$ and $S^{+}_{\lambda,\alpha}(x):=\#\{p_{n+1}\leqslant x\colon p_{n+1}-p_{n}\geqslant\lambda\log^{\alpha}x\}$.
Is it presentely known, 6 years after Yitang Zhang's 2013 breakthrough, whether $S_{1,1-\alpha}^{-}(x)\sim S_{1,1+\alpha}^{+}(x)$? If not, is it a consequence of some widely believed conjecture such as Hardy-Littlewood $k$-tuple conjecture?
